Object Oriented Data Analysis (Chapman & Hall/CRC Monographs on Statistics and Applied Probability)

Description

Object Oriented Data Analysis is a framework that facilitates inter-disciplinary research through new terminology for discussing the often many possible approaches to the analysis of complex data. Such data are naturally arising in a wide variety of areas. This book aims to provide ways of thinking that enable the making of sensible choices. The main points are illustrated with many real data examples, based on the authors' personal experiences, which have motivated the invention of a wide array of analytic methods. While the mathematics go far beyond the usual in statistics (including differential geometry and even topology), the book is aimed at accessibility by graduate students. There is deliberate focus on ideas over mathematical formulas. Cover Half Title Series Page Title Page Copyright Page Dedication Contents Preface 1. What Is OODA? 1.1. Case Study: Curves as Data Objects 1.2. Case Study: Shapes as Data Objects 1.2.1. The Segmentation Challenge 1.2.2. General Shape Representations 1.2.3. Skeletal Shape Representations 1.2.4. Bayes Segmentation via Principal Geodesic Analysis 2. Breadth of OODA 2.1. Amplitude and Phase Data Objects 2.2. Tree-Structured Data Objects 2.3. Sounds as Data Objects 2.4. Images as Data Objects 3. Data Object Definition 3.1. OODA Foundations 3.1.1. OODA Terminology 3.1.2. Object and Feature Space Example 3.1.3. Scree Plots 3.1.4. Formalization of Modes of Variation 3.2. Mathematical Notation 3.3. Overview of Object and Feature Spaces 3.3.1. Example: Probability Distributions as Data Objects 4. Exploratory and Confirmatory Analyses 4.1. Exploratory Analysis–Discover Structure in Data 4.1.1. Example: Tilted Parabolas FDA 4.1.2. Example: Twin Arches FDA 4.1.3. Case Study: Lung Cancer Data 4.1.4. Case Study: Pan-Cancer Data 4.2. Confirmatory Analysis–Is It Really There? 4.3. Further Major Statistical Tasks 5. OODA Preprocessing 5.1. Visualization of Marginal Distributions 5.1.1. Case Study: Spanish Mortality Data 5.1.2. Case Study: Drug Discovery Data 5.2. Standardization–Appropriate Linear Scaling 5.2.1. Example: Two Scale Curve Data 5.2.2. Overview of Standardization 5.3. Transformation–Appropriate Nonlinear Scaling 5.4. Registration–Appropriate Alignment 6. Data Visualization 6.1. Heat-Map Views of Data Matrices 6.2. Curve Views of Matrices and Modes of Variation 6.3. Data Centering and Combined Views 6.4. Scatterplot Matrix Views of Scores 6.5. Alternatives to PCA Directions 7. Distance Based Methods 7.1. Fréchet Centers In Metric Spaces 7.2. Multi-Dimensional Scaling For Object Representation 7.3. Important Distance Examples 7.3.1. Conventional Norms 7.3.2. Wasserstein Distances 7.3.3. Procrustes Distances 7.3.4. Generalized Procrustes Analysis 7.3.5. Covariance Matrix Distances 8. Manifold Data Analysis 8.1. Directional Data 8.2. Introduction to Shape Manifolds 8.3. Statistical Analysis of Shapes 8.4. Landmark Shapes 8.4.1. Shape Tangent Space 8.4.2. Case Study: Digit 3 Data 8.4.3. Case Study: DNA Molecule Data 8.4.4. Principal Nested Shape Spaces 8.4.5. Size-and-shape space 8.4.6. Further Methodology 8.5. Central Limit Theory on Manifolds 8.6. Backwards PCA 8.7. Covariance Matrices as Data Objects 9. FDA Curve Registration 9.1. Fisher-Rao Curve Registration 9.1.1. Example: Shifted Betas Data 9.1.2. Introduction to Warping Functions 9.1.3. Fisher-Rao Mathematics 9.2. Principal Nested Spheres Decomposition 10. Graph Structured Data Objects 10.1. Arterial Trees as Data Objects 10.1.1. Combinatoric Approaches 10.1.2. Phylogenetics 10.1.3. Dyck Path 10.1.4. Persistent Homology 10.1.5. Comparison of Tree Analysis Methods 10.2. Networks as Data Objects 10.2.1. Graph Laplacians 10.2.2. Example: A Tale of Two Cities 10.2.3. Extrinsic and Intrinsic Analysis 10.2.4. Case Study: Corpus Linguistics 10.2.5. Labeled versus Unlabeled Nodes 11. Classification–Supervised Learning 11.1. Classical Methods 11.2. Kernel Methods 11.3. Support Vector Machines 11.4. Distance Weighted Discrimination 11.5. Other Classification Approaches 12. Clustering–Unsupervised Learning 12.1. K-Means Clustering 12.2. Hierarchical Clustering 12.3. Visualization Based Methods 12.3.1. Hybrid Clustering Methods 13. High-Dimensional Inference 13.1. DiProPerm–Two Sample Testing 13.2. Statistical Significance in Clustering 13.2.1. High Dimensional SigClust 14. High Dimensional Asymptotics 14.1. Random Matrix Theory 14.2. High Dimension Low Sample Size 14.3. High Dimension Medium Sample Size 15. Smoothing and SiZer 15.1. Why Not Histograms?–Hidalgo Stamps Data 15.2. Smoothing Basics–Bralower Fossils Data 15.3. Smoothing Parameter Selection 15.4. Statistical Inference and SiZer 15.4.1. Case Study: British Family Incomes Data 15.4.2. Case Study: Bralower Fossils Data 15.4.3. Case Study: Mass Flux Data 15.4.4. Case Study: Kidney Cancer Data 15.4.5. Additional SiZer Applications and Variants 16. Robust Methods 16.1. Robustness Controversies 16.2. Robust Methods for OODA 16.2.1. Case Study: Cornea Curvature Data 16.2.2. Case Study: Genome-Wide Association Data 16.3. Other Robustness Areas 17. PCA Details and Variants 17.1. Viewpoints of PCA 17.1.1. Data Centering 17.1.2. Singular Value Decomposition 17.1.3. Gaussian Likelihood View 17.1.4. PCA Computational Issues 17.2. Two Block Decompositions 17.2.1. Partial Least Squares 17.2.2. Canonical Correlations 17.2.3. Joint and Individual Variation Explained 18. OODA Context and Related Areas 18.1. History and Terminology 18.2. OODA Analogy with Object-Oriented Programming 18.3. Compositional Data Analysis 18.4. Symbolic Data Analysis 18.5. Other Research Areas Bibliography Index